i'm developing my own little Android application (Android Studio). In that app i would like to implement an "add new OSM objects" function e.g. adding nodes, ways or areas and furthermore upload these objects from my app to OSM Server. is it possible at all to upload some data from own developed Android application to OSM server? Do I need any permissions for that? Do you have some suggestions for me for the implementation or some useful links? I downloaded and took a look at the open source code of the Vespucci App but that code is really big and complicated.

You naturally can upload new and edited elements via the Editing API, that is what it is there for. You need an OSM account to write data, there is no special authorization required beyond that.

it is good practice, not to say a requirement, that you provide a facility that allows the user to determine if the object they believe is missing is actually not already mapped, this includes all possible ways the object in question can be created in OSM